 reserve a,x for Real;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,h,f1,f2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem Th44:
  Z c= dom (( #Z 2)*(arccot)) & Z c= ]. -1,1 .[ implies
  -(1/2)(#)(( #Z 2)*(arccot)) is_differentiable_on Z & for x st x in Z holds
  ((-(1/2)(#)(( #Z 2)*(arccot)))`|Z).x = arccot.x / (1+x^2)
proof
   assume
A1:Z c= dom (( #Z 2)*(arccot)) & Z c= ]. -1,1 .[;
then A2:Z c= dom((1/2)(#)(( #Z 2)*(arccot))) by VALUED_1:def 5;
then A3:Z c= dom (-(1/2)(#)(( #Z 2)*(arccot))) by VALUED_1:8;
A4:(1/2)(#)(( #Z 2)*(arccot)) is_differentiable_on Z by A2,A1,SIN_COS9:94;
then A5:(-1)(#)((1/2)(#)(( #Z 2)*(arccot))) is_differentiable_on Z
by A3,FDIFF_1:20;
A6:( #Z 2)*(arccot) is_differentiable_on Z & for x st x in Z holds
    ((( #Z 2)*(arccot))`|Z).x = -2*(arccot.x) #Z (2-1) / (1+x^2)
      by A1,SIN_COS9:92;
    for x st x in Z holds
    ((-(1/2)(#)(( #Z 2)*(arccot)))`|Z).x = arccot.x / (1+x^2)
    proof
      let x;
      assume
A7: x in Z;then
A8:(1/2)(#)(( #Z 2)*(arccot)) is_differentiable_in x by A4,FDIFF_1:9;
A9:( #Z 2)*(arccot) is_differentiable_in x by A6,A7,FDIFF_1:9;
  ((-(1/2)(#)(( #Z 2)*(arccot)))`|Z).x=diff(-(1/2)(#)(( #Z 2)*(arccot)),x)
         by A5,A7,FDIFF_1:def 7
      .=(-1)*(diff((1/2)(#)(( #Z 2)*(arccot)),x)) by A8,FDIFF_1:15
      .=(-1)*((1/2)*diff(( #Z 2)*(arccot),x)) by A9,FDIFF_1:15
      .=(-1)*((1/2)*((( #Z 2)*(arccot))`|Z).x) by A6,A7,FDIFF_1:def 7
      .=(-1)*((1/2)*(-2*(arccot.x) #Z (2-1) / (1+x^2)))
         by A1,A7,SIN_COS9:92
      .=(-1)*(-(1/2)*(2*(arccot.x) #Z 1 / (1+x^2)))
      .=(-1)*(-(1/2)*(2*arccot.x / (1+x^2))) by PREPOWER:35
      .=arccot.x / (1+x^2);
    hence thesis;
   end;
   hence thesis by A3,A4,FDIFF_1:20;
end;
